# American Institute of Mathematical Sciences

January  2020, 40(1): 33-46. doi: 10.3934/dcds.2020002

## Hereditarily non uniformly perfect non-autonomous Julia sets

 1 Department of Mathematics, University of Rhode Island, 5 Lippitt Road, Room 102F, Kingston, RI 02881, USA 2 Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, USA 3 Course of Mathematical Science, Department of Human Coexistence, Graduate School of Human and Environmental Studies, Kyoto University, Yoshida-nihonmatsu-cho, Sakyo-ku, Kyoto 606-8501, Japan

Received  October 2018 Revised  June 2019 Published  October 2019

Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in [19] who gave several examples of such sets based on Cantor set-like constructions using nested intervals. We exhibit a class of examples in non-autonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary. In particular, we give a sharp criterion for when Julia sets from our class will be HNUP and we show that the maximum possible Hausdorff dimension of $1$ for these Julia sets can be attained. The proof of the latter considers the Julia set as the limit set of a non-autonomous conformal iterated function system and we calculate the Hausdorff dimension using a version of Bowen's formula given in the paper by Rempe-Gillen and Urbánski [15].

Citation: Mark Comerford, Rich Stankewitz, Hiroki Sumi. Hereditarily non uniformly perfect non-autonomous Julia sets. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 33-46. doi: 10.3934/dcds.2020002
##### References:
 [1] L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics. [2] Francisco Balibrea, On problems of topological dynamics in non-autonomous discrete systems, Appl. Math. Nonlinear Sci., 1 (2016), 391-404.  doi: 10.21042/AMNS.2016.2.00034. [3] E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, FL, 2008. [4] L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9. [5] M. Comerford, A survey of results in random iteration, Proceedings Symposia in Pure Mathematics, American Mathematical Society, 72 (2004), 435–476. [6] M. Comerford, Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynamical Systems, 26 (2006), 353-377.  doi: 10.1017/S0143385705000441. [7] A. Eremenko, Julia Sets are Uniformly Perfect, Preprint, Purdue University, 1992. [8] K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, third edition, 2014. Mathematical foundations and applications. [9] J. E. Fornæss and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708.  doi: 10.1017/S0143385700006428. [10] A. Hinkkanen, Julia sets of rational functions are uniformly perfect, Math. Proc. Cambridge Philos. Soc., 113 (1993), 543-559.  doi: 10.1017/S0305004100076192. [11] S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., (4) (1996), 205-233. [12] R. Mañé and L. F. da Rocha, Julia sets are uniformly perfect, Proc. Amer. Math. Soc., 116 (1992), 251-257.  doi: 10.1090/S0002-9939-1992-1106180-2. [13] C. T. McMullen, Complex Dynamics and Renormalization, Volume 135 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1994. [14] C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740.  doi: 10.1007/s00039-010-0078-3. [15] L. Rempe-Gillen and M. Urbański, Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc., 368 (2016), 1979-2017.  doi: 10.1090/tran/6490. [16] O. Sester, Hyperbolicité des polynȏmes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428. [17] R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575.  doi: 10.1090/S0002-9939-00-05313-2. [18] R. Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, Ⅱ, Discrete Contin. Dyn. Syst., 32 (2012), 2583-2589.  doi: 10.3934/dcds.2012.32.2583. [19] R. Stankewitz, H. Sumi and T. Sugawa, Hereditarily non uniformly perfect sets, Discrete Contin. Dyn. Syst S, 12 (2019), 2391-2402.  doi: 10.3934/dcdss.2019150. [20] H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019.  doi: 10.1088/0951-7715/13/4/302. [21] H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603.  doi: 10.1017/S0143385701001286. [22] H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922.  doi: 10.1017/S0143385705000532. [23] H. Sumi, Dynamics of postcritically bounded polynomial semigroups Ⅲ: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles, Ergodic Theory Dynam. Systems, 30 (2010), 1869-1902.  doi: 10.1017/S0143385709000923. [24] W. Zhiying, Moran sets and Moran classes, Chinese Sci. Bull., 46 (2001), 1849-1856.  doi: 10.1007/BF02901155.

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##### References:
 [1] L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics. [2] Francisco Balibrea, On problems of topological dynamics in non-autonomous discrete systems, Appl. Math. Nonlinear Sci., 1 (2016), 391-404.  doi: 10.21042/AMNS.2016.2.00034. [3] E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, FL, 2008. [4] L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9. [5] M. Comerford, A survey of results in random iteration, Proceedings Symposia in Pure Mathematics, American Mathematical Society, 72 (2004), 435–476. [6] M. Comerford, Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynamical Systems, 26 (2006), 353-377.  doi: 10.1017/S0143385705000441. [7] A. Eremenko, Julia Sets are Uniformly Perfect, Preprint, Purdue University, 1992. [8] K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, third edition, 2014. Mathematical foundations and applications. [9] J. E. Fornæss and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708.  doi: 10.1017/S0143385700006428. [10] A. Hinkkanen, Julia sets of rational functions are uniformly perfect, Math. Proc. Cambridge Philos. Soc., 113 (1993), 543-559.  doi: 10.1017/S0305004100076192. [11] S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., (4) (1996), 205-233. [12] R. Mañé and L. F. da Rocha, Julia sets are uniformly perfect, Proc. Amer. Math. Soc., 116 (1992), 251-257.  doi: 10.1090/S0002-9939-1992-1106180-2. [13] C. T. McMullen, Complex Dynamics and Renormalization, Volume 135 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1994. [14] C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740.  doi: 10.1007/s00039-010-0078-3. [15] L. Rempe-Gillen and M. Urbański, Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc., 368 (2016), 1979-2017.  doi: 10.1090/tran/6490. [16] O. Sester, Hyperbolicité des polynȏmes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428. [17] R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575.  doi: 10.1090/S0002-9939-00-05313-2. [18] R. Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, Ⅱ, Discrete Contin. Dyn. Syst., 32 (2012), 2583-2589.  doi: 10.3934/dcds.2012.32.2583. [19] R. Stankewitz, H. Sumi and T. Sugawa, Hereditarily non uniformly perfect sets, Discrete Contin. Dyn. Syst S, 12 (2019), 2391-2402.  doi: 10.3934/dcdss.2019150. [20] H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019.  doi: 10.1088/0951-7715/13/4/302. [21] H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603.  doi: 10.1017/S0143385701001286. [22] H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922.  doi: 10.1017/S0143385705000532. [23] H. Sumi, Dynamics of postcritically bounded polynomial semigroups Ⅲ: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles, Ergodic Theory Dynam. Systems, 30 (2010), 1869-1902.  doi: 10.1017/S0143385709000923. [24] W. Zhiying, Moran sets and Moran classes, Chinese Sci. Bull., 46 (2001), 1849-1856.  doi: 10.1007/BF02901155.
How the survival sets ${\mathcal S}_k$ are nested. The pictures show preimages of $\overline {\mathrm D}(0,2)$ at stages $M_k$ (in red) and $M_{k-1}$ (in blue) with $m_k = 3$. The dashed blue circle is ${\mathrm C}(0,2)$ while the unit circle is shown in black. Observe how $Q_{M_{k-1},M_k}^{-1}(\overline {\mathrm D}(0,2)) \subset \overline {\mathrm D}(0, 2) \setminus \overline {\mathrm D}(0, 1)$ as in Remark 1(c) is shown in red at Stage $M_{k-1}$
Schematic for the proof of Theorem 1.6 in the case where $\limsup |c_k| = +\infty$. Note how the round annulus ${\mathrm A}(\sqrt{-c_{k}}, 1, \sqrt{|c_{k}|})$ at stage $M_{k-1} + m_k$ (in this case $M_1+m_2$) is pulled back conformally first by the preimage branches of $Q_{M_{k-1},M_{k-1}+m_k}$ to form half the members of the collection $\mathcal C$ at Stage $M_1$. Then the preimage branches of $Q_{M_{k-1}}$ pull back the annuli in $\mathcal C$ (one of which is visible in the zoomed box) to conformal annuli which separate the components of $\mathcal{S}_k$ at stage $0$
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